CBSE Class 10 Mathematics Real Numbers Notes Set 12

Divisibility and Euclid's Division Lemma

A non-zero integer a divides b when b = a × c for some integer c. Then a is a factor (divisor) of b and b is a multiple of a. Every integer is divisible by 1 and −1, and no integer is divisible by 0.
Euclid's division lemma: for positive integers a and b there are unique integers q and r with a = bq + r and 0 ≤ r < b. If r = 0 then b divides a.

HCF and LCM

The HCF of numbers is the greatest number dividing all of them; the LCM is the smallest number divisible by all of them. HCF is never larger, and LCM is never smaller, than any of the given numbers, and the HCF always divides the LCM.
For two numbers a and b: HCF(a, b) × LCM(a, b) = a × b. This relation holds for two numbers only, not for three.
By prime factorisation: HCF uses the least power of each common prime, and LCM uses the greatest power of every prime that occurs. Coprime numbers have HCF 1 and their LCM is their product.

Euclid's Division Algorithm

To find HCF(a, b) with a > b, write a = bq + r. If r = 0 the HCF is b; otherwise replace (a, b) by (b, r) and repeat. The last non-zero remainder is the HCF. The HCF d can always be written as d = xa + yb for integers x and y, and this form is not unique.

Fundamental Theorem of Arithmetic

Every composite number is a product of primes, and this factorisation is unique apart from the order of the factors. If a prime p divides a², then p divides a. This is used to prove that numbers such as √2, √3 and √5 are irrational.

Decimal Expansion of Rational Numbers

Write a rational number p/q in lowest terms. If the prime factorisation of q has the form 2ᵐ × 5ⁿ, the decimal terminates. Otherwise the decimal is non-terminating and repeating.
Example: 13/325 = 1/25 = 0.04 terminates, but 29/343 does not because 343 = 7³.

Worked Example

Find the HCF of 65 and 117 and write it as 65m + 117n.
117 = 65 × 1 + 52, 65 = 52 × 1 + 13, 52 = 13 × 4. So HCF = 13.
13 = 65 − 52 = 65 − (117 − 65) = 2 × 65 − 1 × 117, so m = 2 and n = −1.